TITLE: Applications of the Mean Value Theorem
QUESTION [18 upvotes]: What are some interesting applications of the Mean Value Theorem for derivatives? Both the 'extended' or 'non-extended' versions as seen here are of interest.
So far I've seen some trivial applications like finding the number of roots of a polynomial equation. What are some more interesting applications of it?
I'm asking this as I'm not exactly sure why MVT is so important - so examples which focus on explaining that would be appreciated.

REPLY [2 votes]: MVT is very important. In calculus and analysis, of course. But it's important in other areas too, like applied mathematics an even number theory. For example, for showing Liouville numbers are transcendental.